Quantum-memory-assisted entropic uncertainty relation with moving quantum memory inside a leaky cavity
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Quantum-memory-assisted entropic uncertainty relation with moving quantum memory inside a leaky cavity Soroush Haselia Faculty of Physics, Urmia University of Technology, Urmia, Iran Received: 24 April 2020 / Accepted: 10 September 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Uncertainty principle has a fundamental role in quantum theory. This principle states that it is not possible to measure two incompatible observers simultaneously. The uncertainty principle is expressed logically in terms of standard deviation of the measured observables. In quantum information, it has been shown that the uncertainty principle can be expressed by means of Shannon’s entropy. Entropic uncertainty relation can be improved by considering an additional particle as the quantum memory. In the presence of quantum memory the entropic uncertainty relation is called quantum-memory-assisted entropic uncertainty relation. In this work, we will consider the case in which the quantum memory moves inside the leaky cavity. We will show that by increasing the velocity of the quantum memory, the entropic uncertainty lower bound is decreased.
1 Introduction In the classical world, the measurement error is due to the inaccuracy of the measuring device, while in quantum theory, it is not possible to measure two incompatible observables simultaneously, even if the measurement instrument is accurate. It is because of the uncertainty principle in quantum theory. The first uncertainty relation was proposed by Heisenberg [1]. The Heisenberg uncertainty relation is related to momentum and position observables which are two incompatible observables. Kennard [2] formalized Heisenberg uncertainty principle as x ˆ pˆ x ≥ h¯ /2, where xˆ and pˆ x are standard deviations of the position xˆ momentum pˆ x , respectively. Robertson [3] and Schrodinger ¨ [4] generalized Heisenberg uncertainty ˆ Based on their results, one has relation for arbitrary two incompatible observables Qˆ and R. the following relation for arbitrary quantum state |ψ ˆ Rˆ ≥ 1 |ψ|[ Q, ˆ R]|ψ|, ˆ Q (1) 2 2 and Rˆ = 2 are the stanˆ ˆ ψ| Rˆ 2 |ψ − ψ| R|ψ where Qˆ = ψ| Qˆ 2 |ψ − ψ| Q|ψ ˆ respectively, and Q, ˆ Rˆ = Qˆ Rˆ − Rˆ Q. ˆ The dard deviations of the observables Qˆ and R, left-hand side (l.h.s) of Eq.(1) is called uncertainty and the right-hand side (r.h.s) is called uncertainty lower bound. This uncertainty lower bound is dependent on quantum state |ψ,
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which can lead to a trivial bound when the commutator has zero expectation value. To avoid this drawback and to consider the notion of uncertainty as the (lack of) knowledge about a measurement outcome, it has been suggested to use the Shannon entropy, in an information theoretical framework, as the measure of uncertainty. The important version of entropic uncertainty relation (EUR) was proposed by Kraus [
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